The implementation of a self-consistent constricted variational density functional theory for the description of excited states

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)] and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π(?) orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.     

The formulation of a self-consistent constricted variational density functional theory for the description of excited states

We outline here a self-consistent approach to the calculation of transition energies within density functional theory. The method is based on constricted variational theory (CV-DFT). It constitutes in the first place an improvement over a previous scheme [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, F. Wang, Chem. Phys. 130 (2009) 154102] in that it includes terms in the variational parameters to any desired order n including n = ∞. For n = 2, CV(n)-DFT is similar to TD-DFT. Adiabatic TD-DFT becomes identical to CV(2)-DFT after the Tamm-Dancoff approximation is applied to both theories. We have termed the new scheme CV(n)-DFT. In the second place, the scheme can be implemented self-consistently, SCF-CV(n)-DFT. The procedure outlined here could also be used to formulate a SCF-CV(n) Hartree-Fock theory. The approach is further kindred to the ΔSCF-DFT procedures predating TD-DFT and we describe how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(n)-DFT.     

Loosely bound states of three particles

Loosely bound states of three particles for particles interacting via short range forces were predicted by Efimov. They are shown to arise naturally in the adiabatic hyperspherical approximation presently used to compute reactions along hyperspherical coordinates. Because of the loosely bound nature of such states, the asymptotic form of the effective hyper-radial potential is critical. A variational wave function is suggested and is shown to reproduce the asymptotic potenial obtained by Efimov. Implications and extensions of the theory are suggested.     

The problem of hole localization in inner-shell states of N2 and CO2 revisited with complete active space self-consistent field approach

Potential energy curves for inner-shell states of nitrogen and carbon dioxide molecules are calculated by inner-shell complete active space self-consistent field (CASSCF) method, which is a protocol, recently proposed, to obtain specifically converged inner-shell states at multiconfigurational level. This is possible since the collapse of the wave function to a low-lying state is avoided by a sequence of constrained optimization in the orbital mixing step. The problem of localization of K-shell states is revisited by calculating their energies at CASSCF level based on both localized and delocalized orbitals. The localized basis presents the best results at this level of calculation. Transition energies are also calculated by perturbation theory, by taking the above mentioned MCSCF function as zeroth order wave function. Values for transition energy are in fairly good agreement with experimental ones. Bond dissociation energies for N(2) are considerably high, which means that these states are strongly bound. Potential curves along ground state normal modes of CO(2) indicate the occurrence of Renner-Teller effect in inner-shell states.     

Computation of the nonhomogeneous equilibrium states of a rigid-rod solution

The nonhomogeneous equilibrium phase behavior of a solution of rigid rods is analyzed for a periodic one-dimensional system. Stable and unstable equilibrium solutions for the distribution function are computed as extrema of the free energy of the system expressed by the nonhomogeneous generalization of Onsager's [Ann. N.Y. Acad. Sci. 51, 627 (1949)] theory, which models interaction between rods on the scale of a single rod length. Biaxial equilibrium solutions are computed in a periodic system by discretizing the Euler-Lagrange nonlinear integral equation by the finite-element method and using Newton's method to solve the resulting set of nonlinear equations. Stable states for isotropic-nematic coexistence are computed in a periodic system rather than the semi-infinite system used in previous calculations. The density and order parameter profiles evolve monotically from the isotropic phase to the nematic phase. Unstable, nonhomogeneous, equilibrium states are also computed for concentrations of rods that exceed the value for spinodal decomposition. These nonhomogeneous states are characterized by combinations of bend, twist, and splay distortions in physical space and correspond to unstable attractors in the dynamic process of isotropic-nematic spinodal decomposition. For large systems, the nonhomogeneous states develop wide, bulklike nematic regions separated by thin regions with sharp gradients in orientation. The free energy formulation was also used to compute the accurate neutral stability curve; this curve shows the limits of applicability of the low-wave-number approximations frequently used in the study of spinodal decomposition.     

Multiconfiguration self-consistent field theory of localized orbitals: The secular problem

In MC SCF theory both the orbitals and the wavefunction expansion coefficients are optimized by minimizing the energy of the system with respect to arbitrary variations of the orbitals and the wavefunction expansion coefficients. This procedure leads to separate equations for the optimal orbitals and to the secular equation for the expansion coefficients which must be solved self-consistently. In a previous paper we discussed the properties of several different choices of localization potential which may be used in the orbital equation. In this paper we derive an alternative native secular equation by making use of the transformation properties of the localized orbitals. This secular equation is considerably simpler than the conventional secular equation and leads to a simplified scheme for the self-consistent solution of the orbital and secular equations.     

Semidirect parallel self-consistent field: the load balancing problem in the input/output intensive self-consistent field iterations

The full capacity of contemporary parallel computers can, in the context of iterative ab initio procedures like, for example, self-consistent field (SCF) and multiconfigurational SCF, only be utilized if the disk and input/output (I/O) capacity are fully exploited before the implementation turns to an integral direct strategy. In a recent report on parallel semidirect SCF http://www.tc.cornell.edu/er/media/1996/collabrate.html, http://www.fp.mcs.anl.gd/grand-challenges/chem/nondirect/index.html it was demonstrated that super-linear speedups are achievable for algorithms that exploit scalable parallel I/O. In the I/O-intensive SCF iterations of this implementation a static load balancing, however, was employed, dictated by the initial iteration in which integral evaluation dominates the central processing unit activity and thus determines the load balancing. In the present paper we present the first implementation in which load balancing is achieved throughout the whole SCF procedure, i.e. also in subsequent iterations. The improved scalability of our new algorithm is demonstrated in some test calculations, for example, for 63-node calculation a speedup of 104 was observed in the computation of the two-electron integral contribution to the Fock matrix.Contribution to the Björn Roos Honorary Issue Acknowledgement.We thank J. Nieplocha for valuable help and making the toolkit (including ChemIO) available to us. R.L. acknowledges the Intelligent Modeling Laboratory and the University of Tokyo for financial support during his stay in Japan.     

The electron in a nuclear field problem for highly excited states

A variant of statement of the problem on highly excited states of an electron (Rydberg states in the limit) in the field of nuclei has been considered. It has been shown that this problem can be reduced to the one-center problem with a variable positive charge. The general solution can be found as a linear combination of basis functions corresponding to the stationary states of the single-center model.     

A critique of a thermodynamic description of hydrophobic aggregation in aqueous solution

The conclusions recently summarised by A. Marmur [J. Am. Chem. Soc. 122 (2000) 2120] concerning hydrophobic aggregation of solutes in aqueous solution are examined. The thermodynamic analysis is critically reviewed and the impact of implicit extrathermodynamic assumptions discussed. These assumptions are questioned and shown to lead to a model for hydrophobic aggregation which is flawed.     

The infrared spectra of trifluoromethanesulphonic acid in different states of aggregation

The infrared spectra of CF₃SO₃H in the gaseous, liquid and solid states, as well as dispersed in a solid Ar matrix, were obtained and analyzed. Association through H-bonds is demonstrated in all phases by a broad OH stretching band. Heating of the gas generates new bands due to the monomer, whereas freezing of the liquid causes a slight shift of the ν_OH band to higher frequencies and a drastic decrease in the intensity of the SO₂, deformation band, suggesting a large redistribution of the H-bonds.     

Determination of equilibrium states in multi-phase systems

Volume corrections in calculation of equilibrium state for multi-phase systems, as an extension from one-phase systems, have been described. The algorithms can handle with one, two, three or more phases of different volumes. Several applications of the algorithms to calculation of equilibrium concentrations of all species in two- and three-phase systems have been presented. Different possibilities of the determination of equilibrium constants from extraction data have been discussed.     

Coupling of radiation,excited states and electron energy distribution function in non equilibrium hydrogen plasmas

Some of the recent efforts in the state-to-state modeling of H/H₂-based plasma are considered, with particular concern to the aspects of self-consistent coupling between the chemical kinetics, electron kinetics and radiation. These aspects are first illustrated in the case of a 0-dimensional model considering both optically thin and optically thick cases in recombining and ionizing regimes. In the second part of the paper, a 1-dimensional extension of the model is applied to study a steady normal shock generated in an H₂/He plasma of interest in atmospheric entry problems. A radiation transfer equation is coupled to the model as well, to analyze the effect of radiation transfer on chemical kinetics and compare to the commonly used thin and thick plasma approximations. The most significant result is the influence of reabsorption on the concentration of excited states, which in turn creates additional structure on the electron energy distribution function through second kind collisions.     

The problem of unphysical states in the theory of intermolecular interactions

It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for < 0.8, implying that the perturbation expansion will diverge for larger values of . The behavior of the energy levels as functions of . is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrödinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = . that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.     

Modelling of director equilibrium states in a nematic cell with relief surface

We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell bounded by two parallel surfaces. One surface is planar and the other one is spatially modulated. The relief of the modulated surface is described by a smooth periodic sine-like function. The orientation of NLC director easy axis is assumed to be homeotropic at one cell surface and planar at the other one. The NLC director anchoring with cell surfaces is assumed to be strong. We consider the case where disclination lines occur in the bulk of NLC above the extrema of the modulated surface. These disclination lines run along the crests and troughs of the surface relief. If the orientation of director at both bounding surfaces is of the same type, then NLC director field is continuous. For both configurations mentioned above (with defects and without defects), we obtain analytical expressions for director distribution in the bulk of NLC in the approximation of planar director deformations. Equilibrium distances from disclination lines to the spatially modulated surface are calculated when the defects occur. The dependences of these equilibrium distances on the period and depth of surface relief and the cell thickness are investigated in detail.